Energy Migration and Formation in Quasi-Two-Dimensional Polymer Films As Revealed by the Time-Resolved Fluorescence Depolarization Measurement

Article abstract of DOI:10.1021/jp9840735

Excitation energy migration and subsequent energy transfer to excimer-forming sites occurring in a quasi-two-dimensional plane of poly(isobutyl methacrylate) Langmuir-Blodgett films were investigated through the time-resolved analyses of anthracene fluorescence. Fluorescence anisotropy decays showed that the energy migration occurred more efficiently as the anthracene fraction increased. A further increase of the anthracene gave rise to the excimer-forming sites acting as energy traps, and finally the anthracene fluorescence was markedly quenched from the energy transfer to the traps. The migration process was quantitatively analyzed by a computer simulation based on the Monte Carlo method, which allowed estimation of the number of hopping and the mean-square displacement of excitons. The initial growth of the mean-square displacement with time became gradually gentle, while the number of hopping linearly increased with time. This result shows that the excitons are not diffusive and tend to stay within a small cluster of chromophores, especially for small chromophore density at short times.

Full text of DOI:10.1021/jp9840735

3402
J. Phys. Chem. A 1999, 103, 3402-3409
Energy Migration and Excimer Formation in Quasi-Two-Dimensional Polymer Films As
Revealed by the Time-Resolved Fluorescence Depolarization Measurement
Nobuhiro Sato, Shinzaburo Ito,* Kazutoshi Sugiura, and Masahide Yamamoto
Department of Polymer Chemistry, Graduate School of Engineering, Kyoto UniVersity, Kyoto 606-8501, Japan
ReceiVed: October 14, 1998; In Final Form: February 12, 1999
Excitation energy migration and subsequent energy transfer to excimer-forming sites occurring in a quasi-
two-dimensional plane of poly(isobutyl methacrylate) Langmuir-Blodgett films were investigated through
the time-resolved analyses of anthracene fluorescence. Fluorescence anisotropy decays showed that the energy
migration occurred more efficiently as the anthracene fraction increased. A further increase of the anthracene
gave rise to the excimer-forming sites acting as energy traps, and finally the anthracene fluorescence was
markedly quenched from the energy transfer to the traps. The migration process was quantitatively analyzed
by a computer simulation based on the Monte Carlo method, which allowed estimation of the number of
hopping and the mean-square displacement of excitons. The initial growth of the mean-square displacement
with time became gradually gentle, while the number of hopping linearly increased with time. This result
shows that the excitons are not diffusive and tend to stay within a small cluster of chromophores, especially
for small chromophore density at short times.
Introduction
in the latter case, the fluorescence decay does not vary with the
migration rate. In practice, however, some energy traps, such
as excimers or impurities, often exist, and then the fluorescence
quenching can be observed as a result of the migration and the
subsequent transfer to the traps. To develop efficient energy
transport systems, it is essential to reduce the trap density,
keeping the high chromophore density.
Electronic excitation energy transfer through dipole-dipole
interaction has been widely studied as a powerful tool for
evaluating nanometer-scale distance because the rate of the
transfer markedly depends on the separation distance between
an energy donor and an acceptor. Moreover, the fluorescence
decay profiles reflect the spatial distribution of the donors and
the acceptors so that they provide structural information on the
system where the chromophores are embedded. Recent interest
in artificially organized molecular assemblies gives more
significance to energy-transfer studies on geometrically restricted
structures, such as Langmuir-Blodgett (LB) films,1-7 mi-
celles,8,9 and the phase-separated structure of block copoly-
mers.10-12
On the other hand, the characterization of the transfer process
in such specific structures is an interesting issue from the
standpoint of photophysics. LB films have been extensively
utilized13-18 to control the chromophore-chromophore distance
by changing the layer separation. They also have a feature that
the space dimensionality of the system can be varied by
changing the number of layers of the built-up film.19 As a
limiting case, a two-dimensional field is obtainable by depositing
only a single monolayer. Thus LB films give a unique feature
for the energy-transfer studies. Polymer LB films, in particular,
have advantages from both applicational and scientific points
of view in (1) mechanical and thermal stability, (2) thinness of
each layer, and (3) uniform and random dispersion of functional
groups. Therefore, numerous studies have been reported on the
energy transfer in polymer LB films.17-20
Fayer and co-workers21-23 developed a theory describing the
survival probability of an excitation being on the initially excited
chromophore even for a case that energy traps exist in the
system; this survival probability is directly related to the
fluorescence anisotropy. They excellently demonstrated24 the
coincidence between their theory and the experimental results;
however, mathematical complexity makes it difficult to be
applied to a variety of systems. As will be described below, we
studied a double layer of LB films, which was a quasi-two-
dimensional system, and therefore, it was hard to apply their
theory directly to our system. Later, Fayer et al. presented more
handy expressions based on the two-particle model, which can
be applied to various geometries.25 They also deal with double
layer systems in the paper, but the well-defined expressions are
given only when the layer separation is sufficiently large or
small relative to the Fo¨rster radius. Since the layer separation
of our system (∼1 nm) was comparable to the Fo¨rster radius
(∼2 nm), we could not utilize their treatment. Moreover, while
the decay of fluorescence anisotropy is well-described, behavior
of migrating excitons cannot be understood from this calculation.
Another effective approach for revealing the characteristics
of the excitation energy migration is a computer simulation.
By using appropriate models, one can analyze photophysical
processes in a variety of systems that are hard to deal with
theoretically and also can visualize the motion of the hopping
exciton. For example, Ohmori et al.6 investigated the energy
transfer in polymer LB films by using a Monte Carlo simulation
based on the Fo¨rster kinetics and clarified the relaxation of the
layer structure, assuming a Gaussian distribution of chro-
mophores.
The energy transfer is classified into two cases: one is
“energy transfer” in a narrow sense, occurring between different
kinds of chromophores, and the other is “energy migration”
between the same kind. In the former case, the time course of
the fluorescence intensity well-characterizes the transfer process;
* To whom correspondence should be addressed. E-mail: sito@polym.
kyoto-u.ac.jp.
10.1021/jp9840735 CCC: $18.00 © 1999 American Chemical Society
Published on Web 04/16/1999

Quasi-Two-Dimensional Polymer Films
J. Phys. Chem. A, Vol. 103, No. 18, 1999 3403
TABLE 1: Anthracene Fraction (f) and Weight-Average
Molecular Weight (Mw) of Sample Polymers
sample
f /%
10-3 Mw
A-0.5
A-1
A-3
A-4
A-6
A-12
A-23
A-33
0.53
1.2
2.7
3.7
6.0
12
23
33
95.6
136
89.9
73.0
59.4
52.1
53.5
43.1
In this study, we prepared a quasi-two-dimensional plane by
the LB method using anthracene-labeled poly(isobutyl meth-
acrylate) and investigated the fluorescence behavior to reveal
the photophysical processes in two dimensions. We also applied
a computer simulation based on the Monte Carlo method to
the results of the fluorescence depolarization measurement in
order to reveal the feature of energy migration in two dimen-
sions.
Experimental Part
Materials. The polymer used was poly(isobutyl methacrylate)
(PiBMA), which forms a stable monolayer at the air/water
interface and has good transferability onto solid substrates.26
As a fluorescent probe, various fractions of anthracene moieties
were introduced into the side chain. Anthracene has the
transition moment along its shorter axis and shows large intrinsic
anisotropy. The anthracene-labeled monomer, 9-(10-methylan-
thryl)methyl methacrylate, was obtained as follows. Commercial
10-methylanthracene-9-carboxyaldehyde (Aldrich) was reduced
with NaBH4 in an ethanol solution, and then the product, 9-(10-
methylanthracene)methanol, was reacted with methacryloyl
chloride in a THF solution. The anthracene-labeled polymer was
synthesized by radical copolymerization of isobutyl methacrylate
and 9-(10-methylanthryl)methyl methacrylate in a benzene
solution with AIBN as an initiator. The polymer obtained was
purified through reprecipitation from benzene into methanol
three times. Table 1 shows the characteristics of the polymers
used. The molecular weights were determined by gel permeation
chromatography (GPC) with reference to polystyrene standards.
The fractions of anthracene units were determined from the UV
absorbance with reference to the molar extinction coefficient
of a reference compound, 9-(10-methylanthracene)methanol,
which was found to be 9200 cm-1 M-1 at 377 nm in a
dichloromethane solution. Poly(vinyl pentanal acetal) (PVPe),
employed for the pre- and overcoating layers, was synthesized
by the acetalization of poly(vinyl alcohol) with 1-pentanal. The
detailed synthetic procedure was reported elsewhere.1
Film Preparation. Quasi-two-dimensional planes were pre-
pared in LB films, the structure of which is depicted in Figure
1. Monolayers were prepared on pure water that was treated
with a water purification system (Barnstead NANO Pure II). A
benzene solution (ca. 0.1 g L-1) of the polymer was dropped
onto the water surface in a Teflon-coated aluminum trough.
Twenty minutes after the solution was spread, the monolayer
was compressed up to a surface pressure of 10 mN m-1 and
deposited by the vertical dipping method onto a hydrophobic
quartz substrate treated with trimethylchlorosilane in advance.
Four layers of PVPe were deposited for pre- and overcoating,
respectively, to reduce the influence of the substrate and the
quenching by oxygen in the air. The adjacent two layers of
PiBMA were deposited as a Y-type film.27 The film preparation
and the measurement of surface pressure-area isotherms were
carried out at 20.0 °C.
Figure 1. Layer structure of sample LB film.
Fluorescence Measurement. Steady-state fluorescence spec-
tra were measured with a fluorescence spectrometer (Hitachi
model 850). Time-resolved measurements were carried out with
a time-correlated single photon counting system. A Ti:sapphire
laser (Spectra Physics model 3950) pumped by an Ar ion laser
(Spectra Physics model 2060) was used as a light source. The
beam after passing through a frequency doubler had a wave-
length of 398 nm at which anthracene has a sufficient absor-
bance. The measurement system was setup as follows. An
s-polarized excitation beam was introduced into a sample film
from the opposite side of a detector with an incidence angle of
approximately 45°, and the fluorescence was observed along
the plane normal of the sample film. The parallel and perpen-
dicular components were alternately obtained by rotating an
analyzer by 90°. Optical filters (SC-42 and B-370) were placed
in front of the detector to select monomer emission of anthracene
around 430 nm. A diffuser to remove a polarization effect was
also placed there. A microchannel-plate photomultiplier
(Hamamatsu R3809U) was used as the detector. The full width
at half-maximum (fwhm) of the response function was 60 ps.
For the time-resolved fluorescence spectroscopy, both excita-
tion and emission lights were made depolarized with diffusers.
An optical filter (SC-41) and a monochromator were installed
in front of an end-on photomultiplier (Hamamatsu R3234). The
response function had a fwhm of 600 ps.
The Fo¨rster radius R0 for anthracene-anthracene was deter-
mined from fluorescence and absorption spectra of a model

3404 J. Phys. Chem. A, Vol. 103, No. 18, 1999
Sato et al.
samples with fractions below 3% showed rather high intensity
below 350 nm, owing to the emission of PVPe, which will be
mentioned later.
Figure 4a shows normalized emission spectra excited at 377
nm. As the fraction increased, the peak at 400-410 nm became
weaker and a broad band around 470 nm became stronger. The
emission at shorter wavelengths whose relative intensity in-
creased with decreasing anthracene fraction was attributed to
that of PVPe used for the pre- and overcoating layers. Although
we employed PVPe as a nonfluorescent polymer, it showed a
weak and broad emission over 400-500 nm whose intensity
was comparable to that of the anthracene fluorescence at small
fractions. The higher intensity at ca. 330 nm in the excitation
spectra mentioned before was also ascribed to the PVPe
emission.
Emission intensity from the sample films increased with
increasing anthracene fraction but then decreased at higher
fractions. Figure 4b represents the emission intensity of the peaks
at ca. 430 nm as a function of plane density of the anthracene
unit, which was measured under the same spectroscopic
condition. Although the intensity increased almost in proportion
to anthracene density at low densities, yet fluorescene was
greatly quenched at densities higher than 0.25 nm-2, the value
of which corresponds to the A-6 sample. This behavior will be
discussed together with the results of the fluorescence decay
measurement.
Figure 2. Surface pressure-area isotherms for the monolayers of the
polymers studied. The increase in anthracene fraction gave rise to the
alteration of the profiles, especially at the large fractions..
To investigate the broad band around 450 nm, we examined
time-resolved spectra. As shown in the normalized spectra of
Figure 5, the relative intensity at 470 nm increased with time
for the large fraction samples; therefore, this band can be
assigned to the emission of anthracene excimers. A broad band
of A-1 at short times is due to the PVPe emission.
Fluorescence Decay. The time course of the total emission
intensity, I(t), can be obtained from the parallel and perpen-
dicular components, denoted by I|(t) and I (t), respectively,
according to the equation below.28
Figure 3. Excitation spectra monitored at 430 nm. These curves are
normalized with the intensity at 377 nm. Since the spectra of samples
from A-4 to A-33 are nearly identical, they are not shown here for
clarity. The emission at wavelengths below 350 nm is ascribed to that
from PVPe used for the pre- and overcoating.
I(t) ) I||(t) + 2I (t)
(1)
Figure 6 shows fluorescence decay curves in a logarithmic
scale; the excitation wavelength was 396 nm. The decay curves
from A-0.5 to A-3 were well-fitted with two-component
exponential functions. Judging from the time-resolved fluores-
cence spectra, the fast decay component with a time constant
of ca. 1.5 ns is due to the PVPe emission. Consequently, the
time constant of the slow component (12.5 ns) corresponds to
the intrinsic lifetime of the anthracene in the LB films, being
close to the lifetime of the anthracene moiety in degassed
solutions. As the anthracene fraction increased, the fraction of
the fast component from PVPe decreased and the decay curves
approached a straight line. However, when the fraction was
larger than 6%, the curve deviated again from a straight line
and did not follow even two-component exponential functions.
This deviation indicates the presence of the excitation energy
transfer to energy traps. As the anthracene fraction became
larger, the quenching occurred more markedly.
compound, 9-(10-methylanthryl)methyl trimethyl acetate in
cyclohexane: R0 ) 2.5 nm.
Results
Monolayer Property. Surface pressure-area isotherms are
shown in Figure 2. At small fractions, the isotherms were similar
to that of the unlabeled polymer reported previously.26 As the
fraction increased, however, the surface pressure of the plateau
region (ca. 16 mN m-1) decreased. Furthermore, above 10% in
fraction, the plateau region disappeared and the slope became
steeper. This indicates that the monolayer becomes less com-
pressible as the anthracene fraction increases. The change of
the monolayer property affected the transferability of the
monolayers onto the solid substrate. While the transfer ratios
were close to unity for A-0.5 through A-12, those for A-23 and
A-33 were quite smaller than unity and indicate incomplete
deposition of these monolayers.
Fluorescence Spectra. Figure 3 shows normalized excitation
spectra monitored at 430 nm. The samples with fractions above
4% showed identical spectra that resemble the absorption
spectrum of anthracene in solutions or in thick films. This is
indicative of no special interaction among the anthracene
moieties at the ground state, such as dimer formation. The
Fluorescence Depolarization. The result of the fluorescence
depolarization measurement is shown in Figure 7 where
fluorescence anisotropy ratio, r(t), is defined by the following
equation.
r(t) ) [I|(t) - I (t)]/[I|(t) + 2I (t)]
(2)
Since the small fraction samples showed an anomalous rise from
PVPe emission in the short time region, we omitted it in the

Quasi-Two-Dimensional Polymer Films
J. Phys. Chem. A, Vol. 103, No. 18, 1999 3405
Figure 4. (a) Fluorescence spectra normalized at 430 nm. Emission
of the anthracene excimer appears around 470 nm at large fractions.
The emission around 400 nm was caused by the PVPe emission. (b)
Emission intensity as a function of plane density of the anthracene unit.
Figure 5. Time-resolved fluorescence spectra. The observed time
ranges are 0-1, 10-11, 20-21, and 30-31 ns. The arrows in the figure
represent this time order. Especially for A-33, the excimer band
becomes prominent as time passes.
following discussion. The anisotropy ratio hardly decayed for
A-0.5 and A-1 but decayed faster as the fraction increased. Since
the anthracene moieties embedded in the LB films are hardly
allowed to rotate, the anisotropy decay is caused only by the
energy migration. Consequently, the anisotropy decays indicate
that the migration occurred more efficiently with increasing
anthracene fraction. For A-23 and A-33, however, the decays
became slower despite the large fractions. There may be two
reasons for this: (1) a contribution of the long-lived anthracene
excimer and (2) incomplete deposition of the monolayers from
the air/water interface.
be transferred to more EFS. As a summary of the above
consideration, we can offer the following qualitative schema.
At fractions below 1%, the excited state of the anthracene
relaxed without interacting with one another. For 3-6%, energy
migration between the anthracene moieties appreciably occurred;
however, the monomer fluorescence was hardly quenched
because the migration distance was short, and there were only
a few EFS. Above 6%, the monomer fluorescence was markedly
quenched by the energy transfer to EFS via the efficient energy
migration among the anthracene moieties.
Analysis of the Anisotropy Decay. The anisotropy decay
curve has a direct relationship with the migration process of
the excitation energy. It is consequently possible to explicate
the mode of the migration in the two-dimensional system by
analyzing the obtained anisotropy decays. The current system
is not exactly two-dimensional because the anthracene moieties
are distributed in two layers; accordingly, we attempted to
simulate the anisotropy decays on the basis of the Monte Carlo
method, employing an exciton-hopping model in the double-
layer system. Since energy trapping by EFS complicates
interpretation of the anisotropy decay data, we focused on the
samples from A-0.5 to A-6 in which the quenching by the traps
was negligible.
Discussion
Qualitative Schema for the Migration and the Quenching.
The monomer fluorescence of the anthracene moieties was
appreciably quenched at fractions above 6%. Because the
emission of the anthracene excimer became noticeable above
6%, this quenching was attributable to the excimer-forming sites
(EFS). Considering the anisotropy decay results, the considerable
quenching of the anthracene fluorescence was due not only to
an increase of EFS but also to a more vigorous energy migration
between the anthracene moieties as the fraction became larger.
The apparent transfer distance of the excitation energy became
longer because of the migration so that the excited state could

3406 J. Phys. Chem. A, Vol. 103, No. 18, 1999
Sato et al.
Figure 6. Fluorescence decay curves observed for the excitation at
396 nm. The fast decay observed at short times for A-0.5 is due to the
PVPe emission. At small fractions, the curves are fitted with one-
component exponential functions, if the short component is neglected.
The longer time constant corresponds to the unquenched lifetime of
the anthracene in the film. For large anthracene fractions, the curves
deviate from a straight line, indicating that the energy transfer to energy
traps occurred.
Figure 7. Time course of the measured anisotropy ratio. A-0.5 exhibits
an anomalous rise at short times because of the PVPe emission. For
the samples with fractions larger than 6%, the efficient energy migration
occurred.
Detailed Procedure of the Simulation. Figure 8 schemati-
cally illustrates the arrangement of the simulation space. In the
beginning, two square regions with a side length of L are placed
parallel to each other with a separation of d, and one of the
squares is set as the xy-plane. Then N chromophores are
distributed as follows. One chromophore with an orientation
vector aligning with the y-axis is located at the origin of the
xy-plane, and the other (N - 1) chromophores are randomly
distributed in the two planes with their transition moment vectors
taking random and isotropic orientations. The chromophore at
the origin is excited initially. The distributed chromophores are
distant enough from one another not to come closer than the
excluding area of the chromophore, ∼0.28 nm2. In this condi-
tion, F, the number density of the chromophores in a single
plane, is given as N/(2L2).
Figure 8. Schematic illustration of the arrangement of the simulation
space and the distribution of the chromophores. The x- and y-
components of the orientation vector correspond to the I| and I ,
respectively.
6
-6 -1
kj ) (3/2)κj2R0 |rj| τ0
(4)
Next, a singlet exciton hops around the chromophores in the
following manner, until it is deactivated. First, the migration
rate from the currently excited chromophore to every other one
is calculated. The orientation factor κj2 of chromophore j with
respect to chromophore i is calculated with the following
equation.25
where R0 is the Fo¨rster radius for anthracene-anthracene and
τ0 is the intrinsic lifetime of the anthracene. Second, the
probability of the migration to chromophore j or that of the
deactivation at chromophore i is determined. The former
probability, pj, is given as
2 2
κj2 ) [Mi‚Mj - 3(Mi‚rj) (Mj‚rj)/|rj| ]
(3)
-1
p ) k /( k + τ
0
)
(5)
∑
j
j
j
while the latter, pd, is given as
where Mi and Mj are the orientation vectors of a unit length
and rj is a position vector of chromophore j with respect to
chromophore i. Then, the migration rate kj is given below.
pd ) τ0-1/( k + τ
)
(6)
-1
0
∑
j

Quasi-Two-Dimensional Polymer Films
J. Phys. Chem. A, Vol. 103, No. 18, 1999 3407
The duration time until a next event (migration or deactivation)
is determined with a random number following an exponential
distribution with a parameter of (∑kj + τ0-1). As long as the
exciton is alive, the probability calculation and the exciton
migration are repeated.
When the deactivation occurs, that is, the fluorescence is
observed, the number of photon counts, c(ti), is increased by 1
at the channel ti corresponding to the elapsed time t. Thus, the
profile of c(ti) represents the fluorescence decay. The number
of exciton hopping, h(ti), and the square displacement from the
origin, l2(ti), are also recorded at ti. The simulated anisotropy,
r°sim(ti), is calculated with eq 2 where I (t) and I|(t) correspond
to the squares of the x- and y-components of the orientation
vector, respectively.
The above procedure (distribution, hopping, and deactivation)
is repeated 104-105 times, and finally, the averaged curves of
c(ti) and r°sim(ti) are obtained. The values of h(ti) and l2(ti) are
also averaged and then normalized with the value of c(ti) because
the magnitude of h(ti) and l2(ti) depends on c(ti).
The initial anisotropy ratio r°sim(0) is unity because the
initially excited chromophore is always set along the y-axis,
but actual initial anisotropy ratio is less than unity because the
excitation probability varies with the orientation of excited
chromophores. For example, the isotropically random orientation
yields 2/5 as the ideal initial anisotropy ratio for linearly
polarized excitation. In this simulation, the excitation probability
is considered as a constant multiplier to r°sim(ti). Thus, simulated
anisotropy decay curves are given as
Figure 9. Comparison of the simulated curves with the experimental
decays. The broken lines in the figure indicate the initial anisotropy r0.
Fine results are obtained with F listed in Table 2 and r0 of 0.33.
TABLE 2: Plane Density (G) of Anthracene in a Single
Layer
102 F/nm-2
sample
simulationa
experimental
A-0.5
A-1
A-3
A-4
A-6
0.78
1.2
1.8
3.0
8.8
2.2
4.7
10
14
25
rsim(ti) ) r0‚r°sim(ti)
(7)
where r0 is the initial anisotropy ratio. In principle, r0 can be
determined theoretically if the orientation of chromophores are
known; in practice, however, the experimental initial anisotropy
ratio is often different from the theoretical one. Therefore, r0
was determined so that the experimental decay curves for
different F values might be best fitted with the same r0 value.
Prior to executing the simulation, we checked its validity in
the following two ways. (1) We applied the simulation for a
three-dimensional system after modifying the distribution of the
chromophores and compared with an experimental anisotropy
decay measured for a spin-coated thick film (ca. 0.5 µm thick)
of A-0.5. By using spectroscopically obtained R0 (Fo¨rster
radius), the simulation curve coincided with the experimental
one. This coincidence not only shows the validness of the
simulation but also indicates that R0 obtained for the model
compound in cyclohexane solution is acceptable for the LB films
as mentioned in Experimental Part. (2) The model was modified
for exactly two dimensions, and the simulated curves were
compared with the theoretical expression developed by Baumann
and Fayer.25 When their theory is applied to our case in which
chromophores were distributed in two dimensions with static
and isotropic orientation, the r°sim(t) under the polarized
excitation parallel to the plane is given by
a The values with which the best fitting results were achieved.
Results of the Simulation. The values used for the simulation
are as follows. Fo¨rster radius R0 was 2.5 nm and the unquenched
lifetime τ0 was 12.5 ns. Despite being determined in a solution,
the value of R0 is acceptable even for the LB film because the
results of the simulation and the experiment agreed for the three-
dimensional film, as described in the previous subsection. The
layer separation d was 1.1 nm, as reported for a PiBMA LB
film.26 The appropriate value of L was harder to determine.
Whereas a small simulation area reduces the load of the
computer used, it restricts the diffusion length of excitons and
may yield different results. We determined accordingly the
appropriate size by testing for various simulation areas and found
that 30-40 nm was adequate for L. The value of F is calculated
from N and L.
Figure 9 shows the simulated decays compared with the
experimental decays. As mentioned in the previous section,
anomalous rises from the PVPe emission were found at short
times for the small fraction samples; accordingly, the data points
of the rises are not plotted for clarity. The best fits to the
experimental curves were achieved by using the values of F
listed in Table 2 and r0 ) 0.33.
Figure 10 shows the time evolution of the number of hopping
h and mean-square displacement l2 obtained with the best fits
of rsim(t). One can see that the number of hopping linearly
increases with both time and the fraction. For A-4, as an
example, the exciton hops ca. 100 times during the lifetime of
the anthracene, 12.5 ns. Similarly, the mean-square displacement
increases with both time and the fraction. However, the rises
with time are not linear: the gradient gradually becomes gentler
at short times. This means that the diffusion of the exciton does
not follow Fick’s law with a constant diffusion coefficient. This
r°sim(t) ) exp[-A(t/τ0)1/3]
(8)
with
2
2/3
A ) FπR0 2-2/3(3/2)1/3 |κ| Γ(2/3)
(9)
2/3
where Γ(2/3) is a gamma function and |κ|
equals 0.7641.
The simulated curves followed well this expression, also
indicating the validness of our simulation model.

3408 J. Phys. Chem. A, Vol. 103, No. 18, 1999
Sato et al.
curves reproduced the experimental decay. Nevertheless, con-
trary to our expectation, the values of F for the best fits become
a little smaller (e.g., 0.025 nm-2 for A-4). The cause for this
may be attributed to the fact that a double layer system is not
exactly two-dimensional. Anyhow, these values are far smaller
than the experimental F value, and thus, the preferential
orientation is not the principal cause for the disagreement
between the simulation and the experiment.
Next, since relaxation of the layer structure was also suspected
of causing this disagreement, we carried out the computation
taking account of the dispersion of the chromophores normal
to the film plane. The F values became closer to the experimental
values; however, the fit of simulated decay curves with the
experimental ones got obviously worse.
Considering other possible reasons, we finally ascribed the
observed slow migration rates to the statistical nonuniformity
of the chromophore distribution. Judging from the fluorescence
excitation spectra, we supposed the statistically uniform distri-
bution of the anthracene chromophores in the two-dimensional
plane. However, owing to the strong hydrophobic interaction
among the aromatic rings, the chromophores tend to gather
together. Although this nonuniformity was not detectable by
spectroscopic observation, it might affect the process of energy
migration. Surface pressure-area isotherms told us that the
monolayers were stiffened at high contents of the chromophore,
at which the monolayers formed excimers. These are experi-
mental indications that the anthracene chromophores are likely
to get close. The simulation at small chromophore fractions
revealed that diffusion of excitons is retarded, owing to the
stagnation in the small clusters of chromophores. Therefore, the
present disagreement suggests the weak aggregation of the
chromophores in the monolayer, resulting in the less efficient
migration compared to that for a uniform distribution.
Figure 10. Time evolution of the number of exciton hopping h (top)
and the square displacement l2 from the origin (bottom), which are
obtained from the best-fit simulation. The value of h increases with
time and fraction. The value of l2 also increased with time and the
fraction, but the gradient with time becomes gradually gentler at short
times. The dashed lines indicate the lifetime of anthracene, 12.5 ns.
In any event, judging from the fact that profiles of the
simulated curves conform to those of the experimental decays,
it is safely said that the present simulation well describes the
feature of quasi-two-dimensional migration of excitation ener-
gies which were characterized by the number of hopping,
diffusion length, and their time evolution.
behavior of diffusing excitons, which were also theoretically
predicted by the Gochanour et al.,24 is understood as follows.
When the chromophores are randomly distributed and the
density of them is low, there is a case that some chromophores
form a small cluster. In this case, an exciton moves around the
chromophores only within the cluster and is hard to travel farther
out of the cluster. For A-4, l2 is 7 nm2 at 12.5 ns; that is, the
exciton diffuses ca. 2.6 nm during the lifetime of anthracene,
which is nearly equal to the Fo¨rster radius: R0 ) 2.5 nm.
As shown in Figure 9, the profiles of the experimental
anisotropy decays are well described by this simulation.
However, the values of F listed in Table 2 are different between
the experiment and the simulation. To explain this discrepancy,
we have to consider the orientation of the chromophores,
because the Fo¨rster kinetics is strongly subject to the orientation
of the transition moment vectors. For another possible manner
of orientation, we considered a case that the moment vectors
are positioned along the surface of a cone with a cone angle of
45° in the quasi-two-dimensional film. For the exact two-
dimensional case, this cone model gives the orientation factor,
κ2, smaller than the random and isotropic orientation; that is,
the value of F for the best-fit simulation becomes larger and
approaches the experimental value. According to this model,
we performed the simulation again, and as a result, the simulated
Conclusion
To clarify the nature of photophysical processes in a quasi-
two-dimensional field, we investigated the fluorescence aniso-
tropy of the anthracene-labeled poly(isobutyl methacrylate)
layers included in LB films as adjacent two layers. The time-
resolved fluorescence depolarization measurement revealed that
the excitation energy migration between the anthracene moieties
became more efficient as the anthracene fraction increased. As
the anthracene fraction increased further, excimer-forming sites
appeared and the excitation energy was trapped there via the
efficient energy migration. A computer simulation based on the
Monte Carlo method was applied to the anisotropy decays to
elucidate the mode of the migration. The simulated anisotropy
decay curves reproduced the profiles of the experimental decays,
and the number of hopping and the square displacement of the
migrating excitons were also obtained. An increase in anthracene
fraction led to the increase in both the number of hopping and
the mean-square displacement. For a given fraction, both are
also increased with time, but the increment of the mean-square
displacement gradually got smaller. This result arises from the
property that the exciton tends to stay in small clusters of the
chromophores, especially for the small plane density.

Quasi-Two-Dimensional Polymer Films
J. Phys. Chem. A, Vol. 103, No. 18, 1999 3409
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Acknowledgment. This work was supported by a Grant-
in-Aid for Scientific Research (B) (No. 09450360) from the
Ministry of Education, Science, Sports and Culture of Japan.
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(28) Equation 1 holds on the assumption of the isotropic orientation of
the transition moment vectors. In this work, we did not directly check this
precondition. However, it is likely that the local relaxation of polymer
segments after the deposition leads to the relaxation of some preferential
orientation. As indirect evidence for this, the fluorescence decays obtained
from eq 1 showed one-component exponential decays at small anthracene
fractions when the influence of the PVPe emission was eliminated.